Which player determines whether the ball is “out” or “in”?

The receiving team (not the team who hit the shot) is responsible for calling the ball. It is usually best to defer to the partner who is closest to the spot the ball bounced. If a ball was not clearly out (there was no visible space between the ball and the line), it should be called "in" as fair play requires giving the benefit of the doubt to your opponents. It is also worth mentioning that the call should be made quickly and not open to long debate or weighing of everyone's opinions. Also, spectators are not to be used as judges or "tiebreakers" in case of disagreement. It is strictly between the players, unless it is at a tournament with officials.